/ _--_/
NASA Technical Memorandum 106266
ICOMP-93-25
/ //] * I _ :/ - /../
Viscous Analysis of Three-Dimensional Rotor
Flows Using a Multigrid Method
A. Arnone
Institute for Computational Mechanics in Propulsion
Lewis Research Center
Cleveland, Ohio
and University of Florence
Florence, Italy
(NASA-TM-106266) VISCOUS
OF THREE-DIMENSIONAL ROTOR
USING A MULTIGRIO METHOD
36 p
ANALYSIS
FLOWS
(NASA)
N94-I1481
Unclas
G3/34 0176741
July 1993
VISCOUS ANALYSIS OF THREE-DIMENSIONAL ROTORFLOWS USING A MULTIGRID METHOD
A. Arnone t
Department of Energy Engineering
University of Florence
Florence, ltaly
and
Institute for Computational Mechanics in Propulsion
Lewis Research Center
Cleveland, Ohio
ABSTRACT
A three-dimensional code for rotating blade-row flow analysis has been
developed. The space discretization uses a cell-centered scheme with eigenvalues
scaling for the artificial dissipation. The computational efficiency of a four-stage
Runge-Kutta scheme is enhanced by using variable coefficients, implicit residual
smoothing, and a full-multigrid method.
An application is presented for the NASA rotor 67 transonic fan. Due to the
blade stagger and twist, a zonal, non-periodic H-type grid is used to minimize the
mesh skewness. The calculation is validated by comparing it with experiments in the
range from the maximum flow rate to a near-stall condition. A detailed study of the
flow structure near peak efficiency and near stall is presented by means of pressure
distribution and particle traces inside boundary layers.
t Assistant Professor
INTRODUCTION
In the last decade Fluid Dynamics has undergone impressive evolution both in the
understanding and in the simulation of flow features. In this process, Computational
Fluid Dynamics (CFD) is playing a more and more important role. Modem
turbomachinery operates under very complex three-dimensional flow conditions, and
further improvement requires detailed knowledge of the flow structure. Particularly,
the need to estimate off-design conditions, secondary flows, and heat transfer forces
us to look at viscous models. The real flow inside a turbomachine is unsteady and
strongly influenced by rotor-stator interactions, wake, and clearance effects. Even if
some important steps have been made in the time-accurate and time-averaged
simulation of entire stages (e.g. Rai, 1987, Rao and Delaney, 1990, Adamczyk et al.,
1990), a steady blade-row analysis should still be considered a basic tool for modem
design. The works of Subramanian and Bozzola (1987), Chima and Yokota (1988),
Choi and Knight (1988), Davis et al. (1988), Hah (1989), Nakahashi et al. (1989),
Weber and Delaney (1991), and Dawes (1991) are some important steps in the
prediction of three-dimensional viscous cascade flows.
In 1988, the University of Florence started a joint project with NASA (ICASE
and ICOMP) on viscous cascade flow simulation. During this research project, the
TRAF2D and TRAF3D codes (TRAnsonic Flow 2D/3D) were developed (Amone et
al. 1988, 1991, 1992). Those codes are capable of solving viscous cascade flows
using H-type or C-type grids and of predicting heat transfer effects. In the present
work, the procedure was extended to the case of rotating blade passages. Particular
attention has been dedicated to important aspects such as minimization of the grid
skewness, accuracy, and computational cost.
As for accuracy, non-periodic C- or H-type grids are stacked in three dimensions.
The removal of periodicity allows the grid to be only slightly distorted even for
cascades having a large camber or a high stagger angle and twist. This allows us to
pick up details of the throat flow with a reasonable number of grid points. A very low
level of artificial dissipation is guaranteed by eigenvalues scaling, which is a three-
dimensional extension of the one proposed by Swanson and Turkel (1987), and
Martinelli and Jameson (1988).
The two-layer eddy-viscosity model of Baldwin and Lomax (1978) is used for the
turbulence closure.
As for efficiency, the Reynolds-averaged Navier-Stokes equations are solved
using a Runge-Kutta scheme in conjunction with accelerating techniques. Variable-
coefficient implicit residual smoothing, as well as the Full-Approximation-Storage
multigrid scheme of Bran& (1979) and Jameson (1983) are used in the TRAF3D
code. Those accelerating strategies are implemented in conjunction with grid
refinement to get a Full Multigrid Method.
The code is validated by studying the NASA rotor 67 transonic fan. This fan has
important viscous and three-dimensional effects and experiments are available in a
wide range of flow rates from the maximum one to near stall. Moreover, this
geometrywasrecentlyproposedas an AGARD test case for code validation (Fottner,1990) and calculations from several authors are available for discussion.
By using the accelerating strategies, accurate, viscous 3D solutions can be
obtained in about half an hour on a modem supercomputer such as a Cray Y-MP.
GOVERNING EQUATIONS
Let p, u, v, w, p, T, E, and H denote respectively density, the absolute velocity
components in the x, y, and z Cartesian directions, pressure, temperature, specific
total energy, and specific total enthalpy. The three-dimensional, unsteady, Reynolds-
averaged Navier-Stokes equations can be written for a rotating blade passage in
conservative form in a curvilinear coordinate system g r/, ( as,
cT(j-' Q) 817 8G 8H 8F,, + OG,, +SH,, +I+--+--+--= -- -- (1)
e_ 8v 8( 8_ 8r_ e(
where the Cartesian system x_v,z is rotating with angular velocity t2 around the x
axis,
i.t!pu ,pE
F _j-l
pU
puU + _._p
pvU + _,,p
pwU +_p
pHU .4-_t P
., G=j -1,
oV
ouV +fix p
pvV + rlyp
,owV + rLp
oHV - rl,p
(2a)
n _j-l
"pW
puw +(_p
pvW +(yp
m, rv +(,pp_w +_p
P, 1 -_'
0
0
p.Ow
-p.f2v
0
(2b)
The contravariant velocity components of eqs. (2) are written as,
_=#,+#.+_ v+_,w
3
V = rl, + r&u + tly v + rl, w
and the transformation metrics are defined by,
_,_ = J(Y,_z¢ -Ycz,Q, _y = J(z,Tx¢ - zcx,7)
_: = J(x,TY¢- xcY,1) , rl,: = d( ycz_ - ycz¢)
fly = J(zcxg - z_x¢), r& = J(x_Y¢ -x_Y¢)
(x=J(y_z,l-zcy,7), (y=J(zcx,7-x_z,Q
rl, =-x, r& - yt rly -zt rL, (t = -xt(,: -Yt(r -z,(_
Xt = O, Yt = --F2Z, Zt ='.OY
where the Jacobian of the transformation J is,
J-_ = xcy_z¢ + x_y¢z¢ + x_ycz_ -x_ycz_ -x_y_z_-x¢y,z¢
The viscous flux terms are assembled in the form,
(3)
(4)
(5)
FV ---_ L1-1'
0
4
GV _ j"l,
0
_ +_ +_
(6)
By --_ j-l,
0
_+_+_
_+_+_
where,
r= =2_ux +_(ux +vy +we)
r,_ =2/_vy +2(u_ +vy +w_)
_-- _--_(u_+v_)
_.-_--u(v_+_)fl,,=ur,=+vrw+w r,= +kT_
flr =u r_ + V r_ + W ry_ + k Ty
fl,=ur= +vr_ +w r., +kT_
(7)
and the Cartesian derivatives of (7) are expressed in terms of ¢-, r/-, and (-derivatives
using the chain rule, i.e.,
(8)
The pressure is obtained from the equation of state,
p =pRr (9)
According to the Stokes hypothesis, 2 is taken to be -2g,'3 and a power law is used to
determine the molecular coefficient of viscosity g as function of temperature. The
eddy-viscosity hypothesis is used to account for the effect of turbulence. The
molecular viscosity # and the molecular thermal conductivity k are replaced with,
/a =/a_ +/a t (10)
_ /ak + /.t (11)
where cp is the specific heat at constant pressure, Pr is the Prandtl number, and the
subscripts l and t refer to laminar and turbulent respectively. The turbulent quantities
#, and Pr t are computed using the two-layer mixing length model of Baldwin and
Lomax (1978). The contribution of the eddy viscosity is computed separately in the
blade-to-blade direction r/and in the spanwise direction _. The inverse of the square
of the wall distances d is then used to compute the resulting eddy viscosity,
(12)
1_, = f(l.t) +(1 -/)(#,)¢ (13)
The transitional criteria of Baldwin and Lomax is adopted on the airfoil surface while
on the end walls, the shear layer is assumed to be fully-turbulent from the inlet
boundary.
SPATIAL DISCRETIZATION
Traditionally, using a finite-volume approach, the governing equations are
discretized in space starting from an integral formulation and without any intermediate
mapping (e.g. Jameson at al., 1981, Ni, 1981, Holmes and Tong, 1985). The
transformation metrics of (4) can be then associated with the projections of the face
6
areas as the contravariant components of (3) can be related to the normal components
of the relative velocity. In the present work, due to the large use of eigenvalues and
curvilinear quantities, we found it more convenient to map the Cartesian space (x,y,z)
in a generalized curvilinear one (¢, r/, g). In the curvilinear system, the equation of
motion (1) can be easily rewritten in integral form by means of Green's theorem and
the metric terms are handled following the standard finite-volume formulation. The
computational domain is divided into hexahedrons and the transformation metrics are
evaluated so that the projected areas of the cell-faces are given by the ratio of the
appropriate metric derivatives to the Jacobian ones, i.e. ¢x/J is the projection onto the
x-axis of a cell face at a fixed ¢ location. A cell-centered scheme is used to store the
flow variables. On each cell face the convective and diffusive fluxes are calculated
after computing the necessary flow quantities at the face center. Those quantities are
obtained by a simple averaging of adjacent cell-center values of the dependent
variables.
BOUNDARY CONDITIONS
In cascade calculations we have four different types of boundaries: inlet, outlet,
solid walls, and periodicity. At the inlet, the presence of boundary layers, on hub and
tip end walls, is accounted for by giving a total pressure and a total temperature
profile whose distribution simulates the experimental one. According to the theory of
characteristics, the flow angles, total pressure, total temperature, and isentropic
relations are used at the subsonic-axial inlet, while the outgoing Riemann invariant is
taken from the interior. At the subsonic-axial outlet, the average value of the static
pressure at the hub is prescribed and the density and components of velocity are
extrapolated together with the circumferential distribution of pressure. The radial
equilibrium equation is used to determine the spanwise distribution of the static
pressure. On the solid walls, the pressure is extrapolated from the interior points, and
the no-slip condition and the temperature condition are used to compute density and
total energy. For the calculations presented in this paper, all the walls have been
assumed to be at a constant temperature equal to the total inlet one.
Cell-centered schemes are generally implemented using phantom cells to handle
the boundaries. The periodicity from blade passage to blade passage is, therefore,
easily overirnposed by setting periodic phantom cell values. On the boundaries where
the grid is not periodic, the phantom cells overlap the real ones. Linear interpolations
are then used to compute the value of the dependent variables in phantom cells.
Even if this approach does not guarantee conservation of mass, momentum, and
energy, no accuracy losses have been experienced unless strong flow gradients occur
along non-periodic grid boundaries with strong differences in mesh size. In the
present work, the number of fine cells to be used for interpolation on a coarse cell
never exceeded three.
The clearanceregion is handledby imposingperiodicity conditionsacrosstheairfoil without any modellization of the blade cross-section.
ARTIFICIAL DISSIPATION
In viscous calculations, dissipating properties are present due to diffusive terms.
Away from the shear layer regions, the physical diffusion is generally not sufficient to
prevent the odd-even point decoupling of centered schemes. Thus, to maintain
stability and to prevent oscillations near shocks or stagnation points, artificial
dissipation terms are also included in the viscous calculations. Equation (1) is written
in semi-discrete form as,
8Q ÷C(Q)-D(Q)--o (14)8t
where the discrete operator C accounts for the physical convective and diffusive
terms, while D is the operator for the artificial dissipation. The artificial dissipation
model used in this paper is basically the one originally introduced by Jameson,
Schmidt, and Turkel (1981). In order to minimize the amount of artificial diffusion
inside the shear layer, the eigenvalues scaling of Martinelli and Jameson (1988), and
Swanson and Turkel (1987) have been used to weight these terms. The quantity
D(Q) in eq. (14) is defined as,
D:Q) =(D_¢ -D_ 4-D_ - D_ 4-D_¢-D})Q (15)
where, for example, in the _ curvilinear coordinates we have,
D2_Q = V¢( A,.,,/z,.J, _://zj.k) AeQ,.j.,
D_cQ = 17¢(A,÷,/zj.k _]/zj.k) A¢ V_AcQ,.j.k
(16)
i,j,k are indices associated with the _,r/,( directions and V¢,A¢ are forward and
backward difference operators in the _ direction. The variable scaling factor A is
defined for the three-dimensional case as,
where,
1
A,+,/zj.k :_[(A¢),.j, k +(A¢),,.j.k ] (17)
A¢ = _¢ 2¢ (18)
8
Thedefinitionof the coefficient • has been extended to the three-dimensional case asfollows,
_=1 +(/7"'71"+(/t"¢_ _
txo txU
+_=1 +/;1"_/° -/-/_)
_.l/2°j,k 2) (23)
where typical values for the constants Kce) and K(_) are 1/2 and 1/64 respectively. For
the remaining directions r/and _, the contribution of dissipation is defined in a similar
way. The computation of the dissipating terms is carried out in each coordinate
direction as the difference between first and third difference operators. Those
operators are set to zero on solid walls in order to reduce the global error on the
conservation property and to prevent the presence of undamped modes (Pulliam,
1986, and Swanson and Turkel, 1988,).
It is important to anticipate now that from the definition of residual of (25),
variable scaling, and time steps of (26,27,28), it results that the artificial dissipation is
scaled with a factor proportional to the ratio between the global time step and the
inviscid time step. Close to solid walls, the grid volume is very small and viscous time
step limitation is dominant. The ratio of the time step over the inviscid one becomes
very small and most of the artificial dissipation is removed.
TIME-STEPPING SCHEME
The system of the differential equation (14) is advanced in time using an explicit
four stage Runge-Kutta scheme until the steady-state solution is reached. A hybrid
scheme is implemented, where, for economy, the viscous terms are evaluated only at
the first stage and then frozen for the remaining stages. If n is the index associated
with time we will write it in the form,
1
(_ o) -- O.
e,,_-eo,+__¢o,+=:R(¢,)
e" =eo'+O.÷,
1 1
a2=-_, a3=-_, a4=1
where the residual R (Q) is defined by,
R(Q)=AtJ[C(Q) -D(Q)]
(24)
(25)
10
Good, high-frequencydampingproperties,importantfor the multigrid process,havebeenobtainedby performingtwo evaluationsof the artificialdissipatingterms,at thefirst and secondstages. It is worthwhile to notice that, in the Runge-Kuttatime-steppingschemes,thesteadystatesolutionis independentof the time step;therefore,this steppingisparticularlyamenableto convergenceaccelerationtechniques.
ACCELERATION TECHNIQUES
In order to reduce the computational cost, four techniques are employed to speed
up convergence to the steady state-solution. These techniques: 1) local time-
stepping; 2) residual smoothing; 3) multigrid; 4) grid refinement; are separately
described in the following.
Local Time-Steppina
For steady state calculations with a time-marching approach, a faster expulsion of
disturbances can be achieved by locally using the maximum available time step. In the
present work the local time step limit At is computed accounting for both the
convective (Ate) and diffusive (At,t) contributions as follows,
(26)
where co is a constant usually taken to be the Courant-Friedrichs-Lewy (CFL)
number. Specifically, for the inviscid and viscous time step we used,
1At - (27)
c 2¢ + 2,7+2_
where y is the specific heat ratio and,
S_ = x:¢ : 2 : 2 + 2 :+y¢+z¢, S_=x_ Yo+zo, S_=x_+Y¢+z¢,2 : (29)
11
K t being a constant whose value has been set equal to 2.5 based on numerical
experiments.
Residual Smoothina
An implicit smoothing of residuals is used to extend the stability limit and the
robustness of the basic scheme. This technique was first introduced by Lerat in 1979
in conjunction with Lax-Wendroff type schemes. Later, in 1983, Jameson
implemented it on the gunge-Kutta stepping scheme. In three dimensions we carried
out the residual smoothing in the form,
(30)
where the residual R includes the contribution of the variable time step and is defined
by (25) and R;- is the residual after a sequence of smoothing in the _,r/, and 6"
directions with coefficients fig , P,7 , and fie. For viscous calculations on highlystretched meshes the variable coefficient formulations of Martinelli and Jameson
(1988) and Swanson and Turkel (1987) have proven to be robust and reliable. In the
present paper, the expression for the variable coefficients fl of (30) has been modifiedto be used in three dimensions as follows,
/ ]}2¢ @ -12¢+27+2¢(ill I ]tCFL 2,7 -1 (31)fl,7 = MAX O, CFL" 24+2,/2¢ _,7
2¢ -1
2¢+27+2¢
where the coefficients _, _y and _z are the ones defined in eqs. (19), and CFL, and
CFL* are the Courant numbers of the smoothed and unsmoothed scheme
respectively. For the hybrid four-stage scheme we used CFL=5, and CFL'=2.5.
12
Multiqrid
This technique was developed in the beginning of the 1970s for the solution of
elliptic problems (Brandt, 1979) and later was extended to time-dependent
formulations (Ni, 1981, and Jameson, 1983). The basic idea is to introduce a
sequence of coarser grids and to use them to speed up the propagation of the fine grid
corrections, resulting in a faster expulsion of disturbances. In this work we used the
Full Approximation Storage (FAS) schemes of Brandt (1979) and Jameson (1983).
Coarser, auxiliary meshes are obtained by doubling the mesh spacing and the
solution is defined on them using a rule which conserves mass, momentum, and
energy,
(j-,(_jo))eh= _(j-,Q )h (32)
where the subscripts refer to the grid spacing, and the sum is over the eight cells
which compose the 2/I grid cell. Note that this definition coincides with the one used
by Jameson when the reciprocal of the Jacobians are replaced with the cell volumes.
To respect the fine grid approximation, forcing functions P are defined on the coarser
grids and added to the governing equations. So, after the initialization of Q2h using
eq.(32), forcing functions P2h are defined as,
P2h = ZRh( Qh) - Rah( _°) ) (33)
and added to the residuals Bah to obtain the value /_h which is then used for the
stepping scheme.
= Ba (Q O + (34)
This procedure is repeated on a succession of coarser grids and the corrections
computed on each coarse grid are transferred back to the finer one by bilinear
interpolations.
A V-type cycle with subiterations is used as a multigrid strategy. The process is
advanced from the fine grid to the coarser one without any intermediate interpolation,
and when the coarser grid is reached, corrections are passed back. One Runge-Kutta
step is performed on the h grid, two on the 2h grid, and three on all the coarser grids.
It is our experience in cascade flow calculations that subiterations increase the
robustness of the multigrid.
For viscous flows with very low Reynolds number or strong separation, it is
important to compute the viscous terms on the coarse grids, too. The turbulent
viscosity is evaluated only on the finest grid level and then interpolated on coarse
grids.
13
On each grid, the boundary conditions are treated in the same way and updated atevery Runge-Kutta stage. For economy, the artificial dissipation model is replaced on
the coarse grids with constant coefficient second-order differences.
The interpolations of the corrections introduce high frequency errors. In order to
prevent those errors from being reflected in the eddy viscosity, turbulent quantifies are
updated after performing the stepping on the fine grid. On coarse grids, the turbulent
viscosity is evaluated by averaging the surrounding fine grid values.
Grid Refinement
A grid refinement strategy is used to provide a cost-effective initialization of the
fine grid solution. This strategy is implemented in conjunction with multigrid to
obtain a Full Multigrid (FMG) procedure. With the FMG method, the solution is
initialized on a coarser grid of the basic grid sequence and iterated a prescribed
number of cycles of the FAS scheme. The solution is then passed, by bilinear
interpolations, onto the next, finer grid and the process is repeated until the finest grid
level is reached. In the present paper we have introduced three levels of refinement
with respectively two, three, and four grids.
COMPUTATIONAL GRID
The three-dimensional grids are obtained by stacking two dimensional grids
generated on blade-to-blade surfaces at constant radii (¢, r/ plane). In order to
minimize the grid skewness, in the blade-to-blade projection, the grids structure can
be chosen on the basis of the blade geometry and flow conditions. Turbine blades are
generally characterized by blunt leading edge and high turn with a subsonic incoming
flow in the relative plane. For these geometries, non-periodic C-type grids, have
proven to be effective (e.g. Amone et al., 1991, 1992). In the case of a compressor,
and particularly for a fan, the leading aspect of the geometry is the high stagger and
twist, while the leading edge is often quite sharp (fig. 1). In addition, the incoming
flow in the relative plane can be supersonic for a large part of the blade span. As
consequence, a C-type structure of the grid may smear too much of the bow shock
away from the leading edge (i.e. Weber and Delaney, 1991). In the present work, a
non-periodic H-type grid was implemented. The removal of mesh periodicity allow
the grid to accommodate highly staggered airfoils with a low level of skewness. To
minimize the undesired interaction between strong flow gradients and non-periodic
boundaries the mesh correspondence is broken before the leading edge and on the
wake, but not inside the blade channel. The inviscid grids are elliptically generated,
controlling the grid spacing and orientation at the wall. Viscous blade-to-blade grids
are then obtained from inviscid grids by adding lines near the wall with the desired
spacing distribution.
14
For highly twisted blades, a low grid skewness in the various spanwise sectionscan also be maintained by adjusting the grid-point distributions on the suction and
pressure sides of the blade.
In the spanwise direction (0 a standard H-type structure is used. Near the hub
and tip endwalls, geometric stretching is used for a specified number of grid points,
after which the spanwise spacing remains constant.
APPLICATION AND DISCUSSIONS
As validation of the procedure that has been described above, the TRAF3D code
was used to study the NASA rotor 67 transonic fan. This first-stage rotor of a two-
stage transonic fan was designed and tested with laser anemometer measurements at
NASA Lewis (Pierzga and Wood, 1985). The rotor has 22 low aspect ratio (1.56)
blades and was designed for a rotational speed of 16043 rpm, with a total pressure
ratio of 1.63 and a mass flow of 33. 25 kg/s. Experiments for simple blade-row code
validation are available for the rotor without inlet guide vanes or downstream stators,
and include detailed data near the peak efficiency and stall conditions. This rotor
geometry has been recently proposed as an AGARD test case (Fottner, 1990) and
several authors (Adamczyk et al. 1991, Chima, 1991, Hah and Reid, 1992, Jennions
and Turner, 1992) have computed this geometry trying to understand the complex
nature of transonic rotor flows. Therefore theoretical predictions from different codes
are also available in the bibliography for comparisons and discussions.
A three-dimensional view of an inviscid grid for the rotor is given in fig. 1. In
previous works by the author (Arnone et al., 1991, 1992), a two- and three-
dimensional grid dependency study was carried out in order to figure out the mesh
requirements necessary to obtain a space-converged calculation, especially for viscous
details such as losses, skin friction and heat transfer. Those results can be
extrapolated to the case of rotating blade passages. Using the algebraic turbulence
model of Baldwin and Lomax (1978), ay + at the wall of about four, with a Reynolds
number of about one million, gives satisfactory results unless heat transfer details are
needed. This y+ value was achieved with a mesh spacing at the wall in the blade-to-
blade direction of 2xlO -4 times the hub axial-chord. In the spanwise direction, due to
the relatively thick inlet boundary layer, the mesh spacing at the wall was fixed at
lx10 -3 times the hub axial-chord. One hundred thirty-seven points were used in the
streamwise direction and 49 in the blade-to-blade and hub-to-tip directions. Sixty-five
points were located on the suction and pressure side of the airfoil. In the spanwise
direction, four cells lie inside the clearance region. Three grid sections at 70%, 30%,
and 10% of the span from the shroud, are shown in fig. 2(b), (c), and (d),
respectively. Figure 2 (a) gives a meridional view of the grid.
Following the approach suggested by Pierzga and Wood (1985) the comparison
between calculations and experiments is carded out using the mass flow rate
15
nondimensionalizedwith the choke flow rate as equivalence criteria. However, goodagreement was also found in terms of absolute quantities. The TRAF3D predicts 34.5
kg/s while 34. 96 kg/s was the measured one. This underestimation of the choke flow
rate of about 1.3% agrees with the viscous prediction of Chima (1991) and Jermions
and Turner (1992). The peak efficiency and near stall conditions correspond to a
nondimensional flow rate of.989 and. 924, respectively.
The inlet boundary layer in the endwall region is accounted for by giving a total
pressure profile. The thickness of the boundary layer is taken from experiments and
the 1/7 power law velocity profile is used to estimate the distribution of total pressure.
The core flow is assumed uniform.
The convergence of the root mean square of the norm of residuals is given in fig.
3. This calculation refers to the near-peak efficiency condition, and requires about 35
minutes on the NASA Lewis Cray Y-MP to achieve a four decades' reduction in the
residuals and corresponds to 100 multigrid cycles on the finest grid level using four
grids. More than 35 minutes were needed only for solutions close to the stall
condition. The global mass flow error has always been found to be less then 10 -3 .
Calculations were performed for 13 different values of the nondimensional flow
rate in order reproduce the operating characteristics of the rotor at the design speed.
Results are summarized in fig. 4 in terms of rotor adiabatic efficiency and rotor total
pressure ratio. A nondimensional value of the mass flow rate equal to about. 91 was
the smaller value for which a steady solution was obtained. Further increase in the exit
pressure would produce tip stall. The prediction of the rotor efficiency is quite good
and agrees with the indication of Jennions and Turner (1992) in the fact that the
abrupt decrease going from peak efficiency to stall conditions is smoothed out in the
calculations. The peak efficiency is predicted at a slightly lower mass flow rate.
Some underestimation in the total pressure ratio across the rotor should be noticed,
but it seems to be evident only when approaching the stall condition. It is believed
that the following aspects could be investigated in order understand and/or address
this problem:
the distribution of the inlet boundary layer has been modelled only in terms of
boundary layer thickness while experiments also show some gradients in the
inviscid core of the inlet flow.
the Baldwin-Lomax turbulence model is not very effective for transonic flow
with strong adverse pressure gradients (Dawes, 1990).
measurements of the machine geometry while running have shown some
deformation not included in the present calculations (Fottner, 1990).
the tip clearance model which has been used is quite simple and clearance flow
becomes important when the stall condition is approached (Adamczyk et al.,
1991, and Jennions and Turner, 1992).
16
The quality of the computedsolutionsis evaluatedby comparingthe spanwisedistributionof circumferentialenergy-averagedthermodynamicquantitiesdownstreamof the rotor to experiments.Thepredicteddistributionof staticpressure,flow angle,total pressure,andtotal temperatureat the rotor exit is comparedto experimentsinfigs. 5 and6 for conditionsnearpeakefficiencyandstallrespectively.The agreementis goodon thewhole. Theradialdistributionof the staticpressureis well reproduced(seefigs. 5 (a) and6 (a)), which indicatesagoodestimationof the globallosses.Theexit angleis up to five degreesoff in thecentralpart of thebladespan(figs. 5 (b) and6 (b)), but this agreeswith the calculationsof Chima(1991)andJennionsandTurner(1992).Fromthe plotsof total pressureandtotal temperatureof figs. 5 and6 (c) and(d) we can see that the overall prediction of the rotor characteristic is wellreproduced.
As well known,the flow patterninsidea rotor like theNASA 67 transonicfan isvery complicated and characterizedby effects such as shock-boundarylayerinteraction,clearanceflow, andthree-dimensionalseparationwith vortices roll up.Oneattemptat interpretingthestructureof the computedflow field canbeanalyzingthe relativeflow on blade-to-bladeandmeridionalsurfaces.Picturesof the limitingstreamlinesinsidetheboundarylayercanbeobtainedbymeansof particletraces.
Blade-to-blade flow
Figures 7 and 8 report the computed and measured relative Mach number
contours at three different locations of the blade span. The agreement with
experiments is qualitatively good and the bow shock is not too smeared away from
the leading edge. It is also interesting to notice that those contours agree very well
with the ones obtained by Jennions and Turner (1991) using a two-equation model for
the turbulence closure.
Near peak efficiency, the shock system has a lambda structure with a bow shock.
The passage shock crosses the blade channel and involves about 30% of the upper
part of the airfoil (see figs. 7, and also 10, and 11). Approaching the stall condition
(fig. 8) the passage shock moves upstream and stands in front of the blade so that the
airfoil pressure side is no longer intercepted (see also figs. 12 and 13).
Blade surface flow pattern
Pressure distribution and a restriction of the particle traces close to the airfoil
surface are used to interpret the flow pattern inside the blade boundary layer. A
schematic of this structure for the peak efficiency condition is depicted in fig. 9. As
discussed by Weber and Delaney (1991) and Hah and Reid (1991), most of the
separation and outward flow is observed on the suction side of the blade. The
passage shock is quite strong in the upper part of the rotor and the losses in axial
momentum result in a rapid turn toward the shroud. The separation due to shock-
boundary layer interaction is evident in figs. 10 and 11. Separation lines are
characterized by flows going toward the line, while where the flow reattaches, the
17
particle traces look like they are going away from the line (see. fig. 9). Such a
situation is very clear in figs. 10 (c) and 13 (d). The passage shock also induces
separation on the blade pressure side but only in the very upper part of the airfoil.
The blow up of fig. 10 (c) shows the abrupt radial migration with separation andreattachment.
In the central part of the blade span, the passage shock loses intensity and the
flow lift off is mostly related to the adverse pressure gradient in the axial direction.
Eventually, near the trailing edge, the flow separates on the suction side (see fig.
1l(b)).Close to the blade root, the flow is strongly influenced by a vortex roll up on the
leading edge. As also pointed out by Chima (1991), particles undergo a high relative
incidence close to the hub, which causes the low-momentum fluid to separate and
migrate radially outward ( see figs. 10 (b) and 11 (b)).
In accordance with the calculations of Weber and Delaney (1991), Chima (1991),
and Jennions and Turner (1991), a separation bubble is observed on the blade suction
side near the hub (figs. 9, 11 (b), and 13 (d)).
The flow structure of the near-stall operating condition is qualitatively similar to
the peak efficiency one previously discussed. Now the passage shock has moved
upstream and the pressure side is shock free (fig. 12 (a)) with basically no radial flow
mixing (fig. 12 (b)). On the contrary, on the suction side, the shock involves most of
the blade span (fig. 13 (a)) and induces a strong outward flow with very clear
separation and reattachment lines (fig. 13 (d)). The vortex roll up on the pressure
side of fig. 10 (d) has now moved upstream and stands in front of the blade leading
edge (fig. 13 (c)). The separation bubble on the suction side near the hub looks
slightly smaller. On the shroud, the bow shock interacts with the casing boundary
layer and the flow separates as depicted in fig. 13 (b).
Clearance flow pattern
A picture of the clearance flow pattern is obtained by plotting the relative Machnumber contours and a restriction of the particle traces on a blade-to-blade surface
midway between the blade tip and the casing. Once again the flow structure looks
very similar to the one computed by Jennions and Turner (1991).
The TRAF3D code predicts two tip vortices which intersect before mid channel.
At peak efficiency (fig. 14 (a) and (b)), the shock system still shows a lambda
structure and interacts with the tip vortices. A first vortex is observed close to the
leading edge and a second leakage vortex forms at an axial location just after the
pressure side shock. The two vortices sum up before crossing the passage shock as
shown in fig. 14. When approaching the stall condition there is interaction between
the leading edge shock separation and the leading edge vortex (Adamczyk at. al
(1991), and Jennions and Turner (1991)). The leading edge vortex is now associated
with the casing separation of the bow shock, while the leakage vortex has moved
upstream. The two vortices interact very soon while going towards the pressure side
18
of the next consecutive blade (fig. 5(b) and (c). Figure 15 (d) indicates a strong link
between the shock system and the tip vortices for this flow condition.
Hub endwall flow pattern
Particle traces in the hub boundary layer are reported for completeness in figs. 16
and 17. The high angle of attack experienced by the flow in this region and
previously discussed is evident. The separation bubble close to the trailing edge
causes a easily visible lack of particles, which, when injected close to the blade, roll up
radially.
CONCLUSIONS
The central-difference, finite-volume scheme with eigenvalues scaling for artificial
dissipation terms, variable-coefficient implicit smoothing, and full multigrid has been
extended to predict three-dimensional rotating blade passages. The procedure has
been validate by comparing it with experiment for the NASA rotor 67 in a wide range
of mass flow rate. With these accelerating strategies, detailed three-dimensional
viscous solutions can be obtained for a reasonable fine grid in about half an hour on a
modem supercomputer.
ACKNOWLEDGEMENTS
The author would like to express his gratitude to ICOMP and NASA for
providing computer time for this work. Thanks are also due to Prof. Sergio S.
Stecco of the University of Florence, for encouraging this work.
REFERENCES
Adamczyk, J. J., Celestina, M. L., Beach, T. A., and Barnett, M.,1990,
"Simulation of Three-Dimensional Viscous Flow Within a Multistage Turbine,"
Transactions of the ASME, Vol. 112, July 1990, pp. 370-376.
Adamczyk, J. J., Celestina, M. L., and Greitzer, E. M., 1991, "The Rule of Tip
Clearance in High-Speed Fan Stall," paper 91-GT-83.
Arnone, A., and Swanson, R. C., 1988 " A Navier-Stokes Solver for Cascade
Flows, "NASA CR No. 181682.
Arnone, A., Liou, M.-S., and Povinelli, L. A.,1991, " Multigrid Calculation of
Three-Dimensional Viscous Cascade Flows, " AIAA paper 91-3238.
Amone, A., Liou, M.-S., and Povinelli, L. A.,1992, "Navier-Stokes Solution of
Transonic Cascade Flow Using Non-Periodic C-Type Grids, " Journal of Propulsion
andPower, vol. 8, n. 2, March-April 1992, pp. 410-417.
Baldwin, B. S., and Lomax, H., 1978 "Thin Layer Approximation and Algebraic
Model for Separated Turbulent Flows, " AIAA paper 78-257.
19
Brandt, A., 1979 " Multi-Level Adaptive Computations in Fluid Dynamics, "AIAA paper 79-1455.
Chima, R. V., and Yokota, J. W., 1988 "Numerical Analysis of Three-
Dimensional Viscous Internal Flows," NASA TM 100878.
Chima, R. V., 1991, " Viscous Three-Dimensional Calculations of Transonic Fan
Performance, " AGARD 77 th Symposium on CFD Techniques for Propulsion
Applications, Paper No. 21.
Choi, D., and Knight, C. J., 1988 " Computation of Three-Dimensional Viscous
Liner Cascade Flows, "AIAA Journal, Vol. 26, No. 12, December, pp. 1477-1482.
Davis, R. L., Hobbs, D. E., and Weingold, H. D., 1988, " Prediction of
Compressor Cascade Performance Using a Navier-Stokes Technique," ASME Journal
of Turbomachinery, Vol. 110, pp. 520-531.
Dawes, W. N., 1990, " A Comparison of Zero and One Equation Turbulence
Modelling for Turbomachinery Calculations, " AS/vIE 90-GT-303.
Dawes, N. W., 1991 " The Simulation of Three-Dimensional Viscous Flow in
Turbomachinery Geometries Using a Solution-Adaptive Unstructured Mesh
Methodology, " ASME paper 91-GT-124.
Fottner, L., 1990, " Test Case for Computation of Internal Flows in Aero Engine
Components," Propulsion and Energetic Panel Working Group 18, AGARD-AR-
275.
Hah, C., 1989 " Numerical Study of Three-Dimensional Flow and Heat Transfer
Near the Endwall of a Turbine Blade Row, " AIAA paper 89-1689,.
Hah, C., and Reid, L., 1992 " A Viscous Flow Study of Shock-Boundary Layer
Interaction, Radial Transport, and Wake Development in a Transonic Compressor, "
ASME 90-GT-303.
Holmes, D., G., and Tong, S., S., 1985, " A Three-Dimensional Euler Solver for
Turbomachinery Blade Rows," Journal of Engineering for Gas Turbines and Power,
April 1985, Vol. 107, pp. 258-264
Jameson, A., Schmidt, W., and Turkel, E., 1981 " Numerical Solutions of the
Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping
Schemes, " AIAA paper 81-1259.
Jameson, A., 1983 " The Evolution of Computational Methods in Aerodynamics,
"£ Appl. Mech., Vol. 50.
Jameson, A., 1983 " Transonic Flow Calculations, " MAE Report 1651, MAE
Department, Princeton University.
Jennions, I. K., and Turner, M. G., 1992 " Three-Dimensional Navier-Stokes
Computations of Transonic Fan Flow Using an Explicit Flow Solver and an Implicit
_:-e Solver, " ASME paper 92-GT-309.
Lerat, A., 1979 " Une Classe de Schemas aux Differences Implicites Pour les
Systemes Hyperboliques de Lois de Conservation, "Comptes Rendus Acad. Sciences
Paris, Vol. 288 A.
20
MartineUi,L. andJameson,A., 1988" Validationof a Multigrid Method for theReynolds Averaged Equations, " AIAA paper 88-0414.
Nakahashi, K., Nozaki, O., Kikuchi, K., and Tamura, A., 1989 " Navier-Stokes
Computations of Two- and Three- Dimensional Cascade Flowfields, "AIAA Journal
of Propulsion andPower, Vol. 5, No. 3, May-June, pp. 320-326.
Ni, R-H, 1981 " A Multiple-Grid Scheme for Solving the Euler Equations, "
AIAA paper 81-1025.
Pierzga, M. J., and Wood, J. R., 1985, " Investigation of the Three-Dimensional
Flow Field Within a Transonic Fan Rotor: Experiment and Analysis, " Transaction of
the ASME, vol. 107, April 1985, pp. 436-449.
Pulliam, T. H., 1986 " Artificial Dissipation Models for the Euler Equations, "
AIAA Journal, Vol. 24, No. 12, December, pp. 1931-1940.
Rai, M. M., 1987, " Unsteady Three-Dimensional Navier-Stokes Simulations of
Rotor Stator Interactions, " AIAA paper 87-2058.
Rao, K. V., and Delaney, R. A., 1990 " Investigation of Unsteady Flow Through
a Transonic Turbine Stage, Part I, Analysis," AIAA paper 90-2408.
Subramanian, S. V., and Bozzola, R., 1987 " Numerical Simulation of Three-
Dimensional Flow Fields in Turbomachinery Blade Rows Using the Compressible
Navier-Stokes Equations, " AIAA paper 87-1314.
Swanson, R. C., and Turkel, E., 1987 " Artificial Dissipation and Central
Difference Schemes for the Euler and Navier-Stokes Equations., " AIAA paper 87-
1107.
Weber, K. F., and Delaney, R. A., 1991 " Viscous Analysis of Three-Dimensional
Turbomachinery Flows on Body Conforming Grids Using an Implicit Solver," ASME
paper 91-GT-205.
21
Fig. 1Three-dimensionalinviscidgrid for theNASA rotor 67transonicfan.
22
a) grid in the meridional
plane
b) grid at 70% spanfrom the shroud
c) grid at 30% spanfrom the shroud
d) grid at 10% spanfrom the shroud
Fig. 2 137x49x49 viscous grid for the NASA rotor 67transonic fan.
23
0NASA Rotor 67 near peak efficiency
-I
-2
..-.-3_9
-4O
o
-5
-6
-7
__ I I I I I
0 200 400 600 800 1000
Cycles
Fig. 3 Convergence history for the near peak efficiency
condition (NASA rotor 67 transonic fan).
24
0.98
Z 0.96r_
0.94
r_ 0.92m
0.90
"¢ 0.88
0.86
0.84o[- 0.82o
0.800.90
near P.E.
near Stall (_)
I I I I I
0.92 0.94 0.96 0.98 1.00
1.9
1.8
0f-< 1.7
n_1.6
_D
m 1.5
o 1.4E--
0 1.3E--0n_
1.2
near Stall
.
• computed (i)0 Oexperiment
1.1 _ t _ _ J0.90 0.92 0.94 0.96 0.98 1.00
MASS FLOW RATE/MASS FLOW RATE AT CHOKE
Fig. 4 NASA rotor 67 performance at design speed.
25
a)
e.
1.4
1.3
1.2
I.I
I.OC
0.9
0.80.0
7O
•-----4 computed
© © experiment
I I I I
0.2 0.4 0.6 0.8 1.0
60
< 50
o,,.q
X40
b)30
0.0
2.0
1.8
°1.6Q.
ff 1.4
1.2c)
1.00.0
1.25
1.20
o 1.15E-
_-T 1.10
1.05 ,,-d)
1.000.0
0.2 0.4 0.6 0.8 1.0
p--=_--©-- 0 C :-_
' I I I
0.2 0.4 0.6 0.8 .0
I I I I
0.2 0.4 0.6 0.8 1.0
(R- l_hub)/(Rti p- Rhub)
Fig. 5 Predicted and measured exit survey data near peak
efficiency (NASA rotor 67).
26
a)
b)
c)
d)
1.5
2D. 14
1.3
1.2
1"11
1.O _ computed
0.9 0 0 experiment
0.8 _ z , i0.0 0.2 0.4 0.6 0.8 1.0
70
60'
< 50
xa_ 40
30 I I I 1
0.0 0.2 0.4 0.6 0.8
2.0
1.0
1.8 -
21.6_ ':_--r:_: Q
a._ 1.4
1.2
1.00.0 0.2
1.30
I I I
0.4 0.6 0.8 .0
1.25
o 1.20[-..
%1.15E-- 1.10
1.05
1.000.0
.... _- f'_ z-_ C_. -4_ -_--_
I I I I
0.2 0.4 0.6 0.8 1.0
(R - Rhub)/(Rti p- Rhub)
Fig. 6 Predicted and measured exit survey data near stall
(NASA rotor 67).
27
FLOM
grid at 10% spanfi'om the shroud
/
-1.15
i. •
1.7
/ ._t_,_,o ) grid at 30% span, ',,. from the shroud
FiOII
/ _g_.,J/-illir _ o_,0," "_.-_-_-------------_1grid at 70% span _\_
/', _,\_'i_/r" I _om the shroud _"_FLOU tOlA11q_
Fig. 7 Measured (left) and predicted (right) relative Machnumber contours near peak efficiency (NASA rotor 67).
28
Fig. $ Measured (left) and predicted (right) relative Machnumber contours near stall (NASA rotor 67).
29
clearance flow
passage shock andseparation line
re.attachment line
flow
lift off
trailing-edge separationnear shroud
Fig. 9 Schematic of the suction side boundary-layer flowstructure near peak efficiency (NASA rotor 67)
suction side shock
Fig. 10
a) pressure b) particle traces
Pressure contours and panicle traces close to the blade
pressure side near peak efficiency (NASA rotor 67).
3O
pressure side shock
a) pressure b) particle traces
Fig. 11 Pressure contours and particle traces close to the blade
suction side near peak efficiency (NASA rotor 67)
Fig. 12
suction side shock
a) pressure b) particle traces
Pressure contours and particle traces close to the blade
pressure side near stall (NASA rotor 67).
31
bow shock
a) pressure d)
particle traces
Fig. 13 Pressure contours and particle traces close to the bladesuction side near stall (NASA rotor 67).
leading edge vortex
leakage vortex
a)
passage shock
Fig. 14 Particle traces and relative Mach number contours in the
clearance region near peak efficiency (NASA rotor 67).
32
tip vortex
a)
Fig. 15 Particle traces and relative Mach number contours
in the clearance region near stall (NASA rotor 67).
33
Fig. 16 Panicle traces close to the hub endwall
near peak efficiency (NASA rotor 67)
Fig. 17 Panicle traces close to the hub endwall
near stall (NASA rotor 67)
34
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July 1993
4. TITLE AND SUBTITLE
Viscous Analysis of Three-Dimensional Rotor Flows Using a Multigrid Method
6. AUTHOR(S)
A. Amone
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
National Aeronautics and Space Administration
Lewis Research Center
Cleveland, Ohio 44135-3191
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)
National Aeronautics and Space Administration
Washington, D.C. 20546-0001
Technical Memorandum
5. FUNDING NUMBERS
WU-505-90-5K
PERFORMING ORGANIZATIONREPORT NUMBER
E-7993
10. SPONSORING/MONITORINGAGENCY REPORT NUMBER
NASA TM- 106266
ICOMP-93-25
11. SUPPLEMENTARY NOTES
A. Arnone, Institute for Computational Mechanics in Propulsion, NASA Lewis Research Center and University of
Florence, Department of Energy Engineering, Florence, Italy, (work funded under NASA Cooperative Agreement
NCC3-233). ICOMP Program Director, Louis A. Povinelli, (216) 433-5818.
12a. DISTRIBUTION/AVAILABILITY STATEMENT
Unclassified - Unlimited
Subject Category 34
12b. DISTRIBUTION CODE
13. ABSTRACT (Maximum 200 words)
A three-dimensional code for rotating blade-row flow analysis has been developed. The space discretization uses a
cell-centered scheme with eigenvalues scaling for the artificial dissipation. The computational efficiency of a four-stage
Runge-Kutta scheme is enhanced by using variable coefficients, implicit residual smoothing, and a full-multigrid
method. An application is presented for the NASA rotor 67 transonic fan. Due to the blade stagger and twist, a zonal,
non-periodic H-type grid is used to minimize the mesh skewness. The calculation is validated by comparing it with
experiments in the range from the maximum flow rate to a near-stall condition. A detailed study of the flow structure
near peak efficiency and near stall is presented by means of pressure distribution and particle traces inside
boundary layers.
14. SUBJECT TERMS
Multigrid methods; 3-D rotor flow; Cascades; Navier-Stokes equations
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